Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY CALCULUS

a b c

f(x)

x

Figure 2.5 The graph of a functionf(x), showing that iff(a)=f(c)thenat

one point at least betweenx=aandx=cthe graph has zero gradient.

a b c

C

f(a) A

f(x)

x

f(c)

Figure 2.6 The graph of a functionf(x); at some pointx=bit has the same

gradient as the lineAC.

Mean value theorem

The mean value theorem (figure 2.6) states that if a functionf(x) is continuous

in the rangea≤x≤cand differentiable in the rangea<x<cthen

f′(b)=

f(c)−f(a)

c−a

, (2.20)

for at least one valuebwherea<b<c. Thus the mean value theorem states

that for a well-behaved function the gradient of the line joining two points on the

curve is equal to the slope of the tangent to the curve for at least one intervening

point.

The proof of the mean value theorem is found by examination of figure 2.6, as

follows. The equation of the lineACis

g(x)=f(a)+(x−a)

f(c)−f(a)

c−a

,